Continuity of partial derivatives

[Shaybay92]

What exactly does it mean for a function to have continuous partial derivatives? How do we see this?

[Landau]

Well, first of all the partial derivative(s) must exist. This is a function

x\mapsto D_if(x)

(where D_if denote the i-th partial derivative), and this function itself may be continuous.

If all its partial derivatives exist and are continuous, then the function is differentiable, in the sense that its total derivative exists.